Directed homotopy and homology theory, with an eye towards applications

Directed spaces, i.e., topological spaces equipped with a “sense of direction” of some sort, arise naturally in applications of topology, in particular in computer science and in neuroscience, where they play an increasingly important role.  It is natural to want to develop appropriate directed versions of familiar homotopy invariants, which turns out to be significantly harder than one might expect. 
For example, it is not clear how to formulate a good definition of “directed” homology, even when restricting to directed spaces built from simplices or cubes. To be of theoretical interest, directed homology should be an invariant of an acceptable notion of weak equivalence of directed spaces and should distinguish between at least some 
 directed spaces with the same underlying undirected space. To be of practical interest for applications, it should also be reasonably computable.

• L. Fajstrup, E. Goubault, E. Haucourt, S. Mimram, M. Raussen, Directed Algebraic Topology and Concurrency, Springer Verlag, 2016. 

• M. Grandis, Directed Algebraic Topology. Models of Non-reversible worlds. Cam- bridge University Press, 2010.
Available from Grandis’ website
http://www.dima.unige.it/∼ grandis/BkDAT page.html 

• M. J. Dubut, “Directed homotopy and homology theories of geometric models of true concurrency” (2017 PhD thesis): http://www.lsv.fr/∼dubut/manuscript.pdf 


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